quadratic reciprocity

alternative law of quadratic reciprocity

1. uncountablemathematicsThe mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p.

   Yes, but we need not only Quadratic Reciprocity but also Dirichlet's theorem on primes in arithmetic progressions to see this.

   2007, Paul B. Garrett, Abstract Algebra, Taylor & Francis (Chapman Hall/CRC Press), page 287

   2 more examples

   Gauss studied these sorts of numbers while attempting to formulate and prove higher order reciprocity laws, following his success with quadratic reciprocity.

   2009, Sam Vandervelde, Circle in a Box, American Mathematical Society, page 153

   An interesting consequence of the above algorithm is that one can evaluate the Jacobi symbol in deterministic polynomial time in certain cases analogous to the way (“reduce and flip”) that one computes this symbol using quadratic reciprocity in the case F#61;#92;mathbb#123;Q#125;.

   2013, J. Voight, “Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms”, in Krishnaswami Alladi, Manjul Bhargava, David Savitt, Pham Huu Tiep, editors, Quadratic and Higher Degree Forms, Springer, page 284